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Properties

Label 65.3.q.a
Level $65$
Weight $3$
Character orbit 65.q
Analytic conductor $1.771$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [65,3,Mod(3,65)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(65, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("65.3");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 65.q (of order \(12\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.77112171834\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 6 q^{2} - 2 q^{3} - 16 q^{5} - 4 q^{6} - 2 q^{7} - 72 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 6 q^{2} - 2 q^{3} - 16 q^{5} - 4 q^{6} - 2 q^{7} - 72 q^{8} - 36 q^{11} - 76 q^{12} + 6 q^{13} - 2 q^{15} + 68 q^{16} + 4 q^{17} + 196 q^{18} + 78 q^{20} - 8 q^{21} - 72 q^{22} + 42 q^{23} + 76 q^{25} + 84 q^{26} + 4 q^{27} - 58 q^{28} - 172 q^{30} - 208 q^{31} + 228 q^{32} - 70 q^{33} + 64 q^{35} - 300 q^{36} + 76 q^{37} - 168 q^{38} - 508 q^{40} + 148 q^{41} + 188 q^{42} + 142 q^{43} - 52 q^{45} + 156 q^{46} - 32 q^{47} - 92 q^{48} - 116 q^{50} + 96 q^{51} - 218 q^{52} + 404 q^{53} + 66 q^{55} - 84 q^{56} + 140 q^{57} - 10 q^{58} + 1368 q^{60} - 264 q^{61} + 256 q^{62} - 204 q^{63} + 264 q^{65} + 1232 q^{66} - 218 q^{67} - 304 q^{68} - 108 q^{70} - 480 q^{71} - 486 q^{72} - 820 q^{73} - 360 q^{75} - 436 q^{76} - 140 q^{77} - 396 q^{78} + 168 q^{80} - 444 q^{81} + 286 q^{82} + 752 q^{83} - 258 q^{85} - 552 q^{86} + 266 q^{87} + 390 q^{88} - 368 q^{90} + 500 q^{91} - 1844 q^{92} - 136 q^{93} - 248 q^{95} - 192 q^{96} + 750 q^{97} - 130 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −3.64069 − 0.975519i 0.851395 − 3.17745i 8.83887 + 5.10312i 4.76687 − 1.50897i −6.19933 + 10.7376i −1.04187 − 3.88830i −16.5407 − 16.5407i −1.57708 − 0.910530i −18.8267 + 0.843516i
3.2 −3.48056 − 0.932614i −1.37646 + 5.13702i 7.78045 + 4.49204i −3.98599 − 3.01859i 9.58171 − 16.5960i 0.245535 + 0.916350i −12.6992 − 12.6992i −16.7001 − 9.64179i 11.0583 + 14.2238i
3.3 −2.55155 − 0.683685i 0.0299728 − 0.111860i 2.57887 + 1.48891i −1.53598 + 4.75823i −0.152954 + 0.264924i −0.433652 − 1.61841i 1.90928 + 1.90928i 7.78261 + 4.49329i 7.17226 − 11.0907i
3.4 −1.82787 − 0.489775i 1.34864 − 5.03319i −0.362887 − 0.209513i −4.99967 + 0.0573135i −4.93027 + 8.53948i 1.80409 + 6.73294i 5.91306 + 5.91306i −15.7200 − 9.07594i 9.16680 + 2.34395i
3.5 −1.82159 − 0.488094i −0.458515 + 1.71120i −0.384142 − 0.221785i 4.59261 − 1.97686i 1.67045 − 2.89331i 2.50282 + 9.34064i 5.92549 + 5.92549i 5.07625 + 2.93078i −9.33075 + 1.35940i
3.6 −0.683551 − 0.183157i −0.110205 + 0.411289i −3.03041 − 1.74961i −1.71414 − 4.69699i 0.150661 − 0.260952i −2.42809 − 9.06176i 3.75256 + 3.75256i 7.63722 + 4.40935i 0.311417 + 3.52459i
3.7 0.195436 + 0.0523670i 0.990947 − 3.69826i −3.42865 − 1.97953i 3.54964 + 3.52137i 0.387334 − 0.670883i −2.69566 − 10.0604i −1.13870 − 1.13870i −4.90096 − 2.82957i 0.509325 + 0.874089i
3.8 0.203236 + 0.0544569i −1.01411 + 3.78469i −3.42576 − 1.97786i −3.78794 + 3.26367i −0.412205 + 0.713961i 1.23676 + 4.61567i −1.18365 − 1.18365i −5.50126 − 3.17615i −0.947575 + 0.457015i
3.9 1.45074 + 0.388724i 0.793233 − 2.96038i −1.51057 − 0.872127i 1.16312 − 4.86283i 2.30154 − 3.98639i 2.50910 + 9.36408i −6.10048 − 6.10048i −0.340428 − 0.196546i 3.57769 − 6.60256i
3.10 2.06606 + 0.553599i −0.614105 + 2.29187i 0.498035 + 0.287541i 4.43775 + 2.30355i −2.53756 + 4.39518i −0.00917272 − 0.0342330i −5.18006 − 5.18006i 2.91868 + 1.68510i 7.89343 + 7.21601i
3.11 2.86834 + 0.768571i 0.767334 − 2.86373i 4.17260 + 2.40905i −3.72884 + 3.33103i 4.40196 − 7.62441i 0.227219 + 0.847991i 1.71783 + 1.71783i 0.182090 + 0.105129i −13.2557 + 6.68866i
3.12 3.12392 + 0.837051i −0.842107 + 3.14279i 5.59410 + 3.22975i −2.75743 − 4.17092i −5.26134 + 9.11291i −1.55105 − 5.78859i 5.62456 + 5.62456i −1.37373 − 0.793123i −5.12270 − 15.3377i
22.1 −3.64069 + 0.975519i 0.851395 + 3.17745i 8.83887 − 5.10312i 4.76687 + 1.50897i −6.19933 − 10.7376i −1.04187 + 3.88830i −16.5407 + 16.5407i −1.57708 + 0.910530i −18.8267 − 0.843516i
22.2 −3.48056 + 0.932614i −1.37646 − 5.13702i 7.78045 − 4.49204i −3.98599 + 3.01859i 9.58171 + 16.5960i 0.245535 − 0.916350i −12.6992 + 12.6992i −16.7001 + 9.64179i 11.0583 − 14.2238i
22.3 −2.55155 + 0.683685i 0.0299728 + 0.111860i 2.57887 − 1.48891i −1.53598 − 4.75823i −0.152954 − 0.264924i −0.433652 + 1.61841i 1.90928 − 1.90928i 7.78261 − 4.49329i 7.17226 + 11.0907i
22.4 −1.82787 + 0.489775i 1.34864 + 5.03319i −0.362887 + 0.209513i −4.99967 − 0.0573135i −4.93027 − 8.53948i 1.80409 − 6.73294i 5.91306 − 5.91306i −15.7200 + 9.07594i 9.16680 − 2.34395i
22.5 −1.82159 + 0.488094i −0.458515 − 1.71120i −0.384142 + 0.221785i 4.59261 + 1.97686i 1.67045 + 2.89331i 2.50282 − 9.34064i 5.92549 − 5.92549i 5.07625 − 2.93078i −9.33075 − 1.35940i
22.6 −0.683551 + 0.183157i −0.110205 − 0.411289i −3.03041 + 1.74961i −1.71414 + 4.69699i 0.150661 + 0.260952i −2.42809 + 9.06176i 3.75256 − 3.75256i 7.63722 − 4.40935i 0.311417 − 3.52459i
22.7 0.195436 − 0.0523670i 0.990947 + 3.69826i −3.42865 + 1.97953i 3.54964 − 3.52137i 0.387334 + 0.670883i −2.69566 + 10.0604i −1.13870 + 1.13870i −4.90096 + 2.82957i 0.509325 − 0.874089i
22.8 0.203236 − 0.0544569i −1.01411 − 3.78469i −3.42576 + 1.97786i −3.78794 − 3.26367i −0.412205 − 0.713961i 1.23676 − 4.61567i −1.18365 + 1.18365i −5.50126 + 3.17615i −0.947575 − 0.457015i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
13.c even 3 1 inner
65.q odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 65.3.q.a ✓ 48
5.b even 2 1 325.3.u.b 48
5.c odd 4 1 inner 65.3.q.a ✓ 48
5.c odd 4 1 325.3.u.b 48
13.c even 3 1 inner 65.3.q.a ✓ 48
65.n even 6 1 325.3.u.b 48
65.q odd 12 1 inner 65.3.q.a ✓ 48
65.q odd 12 1 325.3.u.b 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.3.q.a ✓ 48 1.a even 1 1 trivial
65.3.q.a ✓ 48 5.c odd 4 1 inner
65.3.q.a ✓ 48 13.c even 3 1 inner
65.3.q.a ✓ 48 65.q odd 12 1 inner
325.3.u.b 48 5.b even 2 1
325.3.u.b 48 5.c odd 4 1
325.3.u.b 48 65.n even 6 1
325.3.u.b 48 65.q odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(65, [\chi])\).